# Questions tagged [approximations]

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### Is $\pi^2 \approx g$ a coincidence?

In spite of their different dimensions, the numerical values of $\pi^2$ and $g$ in SI units are surprisingly similar, $$\frac{\pi^2}{g}\approx 1.00642$$ After some searching, I thought that this ...
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### How is the Saddle point approximation used in physics?

I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative ...
2k views

### What different approximations yield Gravitoelectromagnetism and Weak Field Einstein Equations?

This question is inspired by this answer, which cites Gravitoelectromagnetism (GEM) as a valid approximation to the Einstein Field Equations (EFE). The wonted presentation of gravitational waves is ...
5k views

### Why does a simple pendulum or a spring-mass system show simple harmonic motion only for small amplitudes?

I've been taught that in a simple pendulum, for small $x$, $\sin x \approx x$. We then derive the formula for the time period of the pendulum. But I still don't understand the Physics behind it. Also, ...
3k views

### Far away from a charged conductor, the field is like a point charge. Where's the point located?

In the framework of classical electrodynamics, at distances much greater than a conductor's dimension, the field ought to approach that of a point charge located at the conductor. But where at? For ...
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I have encountered $\gg$ in many physics text books where it's used as a relation between constants or functions but in none of the text books I have read is it properly defined anywhere. If A \gg ... 1answer 2k views ### Is the existence of a sole particle in an hypothetical infinite empty space explicitly forbidden by QM? Suppose the universe is completely empty with one sole particle trapped in it. To simplify, I will only be looking at the one dimensional case. However, all arguments are applicable for three ... 8answers 3k views ### Why do physicists believe that particles are pointlike? String theory gives physicists reason to believe that particles are 1-dimensional strings because the theory has a purpose - unifying gravity with the gauge theories. So why is it that it's popular ... 4answers 2k views ### The reasoning behind doing series expansions and approximating functions in physics It is usual in physics, that when we have a variable that is very small or very large we do a power series expansion of the function of that variable, and eliminate the high order terms, but my ... 5answers 4k views ### Elliptical Trajectory, or Parabolic? Discuss whether this statement is correct: “In the absence of air resistance, the trajectory of a projectile thrown near the earth’s surface is an ellipse, not a parabola.” Is the above statement ... 5answers 642 views ### Mean field theory Vs Gaussian Approximation? I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the ... 1answer 1k views ### Could Navier-Stokes equation be derived directly from Boltzmann equation? I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ... 3answers 2k views ### Why is a particle non-relativistic when its kinetic energy is small compared to its rest energy? For example, nucleons in nucleus are in motion with kinetic energies of 10 MeV. Their rest energies are about 1000 MeV. Kinetic energy of nucleons is small compared to rest energy. They are hence ... 6answers 3k views ### If the solar system is a non-inertial frame, why can Newton's Laws predict motion? Since there is no object in the universe that doesn't move, and the solar system likely accelerates through space, how did Newton's Laws work so well? Didn't he assume that the sun is the acceleration-... 2answers 3k views ### Why is the Newtonian expression for kinetic energy called the “first order” approximation of the relativistic expression? In many texts, the non-relativistic (Newtonian) kinetic energy formula $$\text{KE}_\text{Newton} =\frac{1}{2}mv^2$$ is referred to as a first order approximation of the relativistic kinetic energy $$\... 5answers 1k views ### Near Earth vs Newtonian gravitational potential Newton's Law of Universal Gravitation tells us that the potential energy of object in a gravitational field is$$U ~=~ -\frac{GMm}{r}.\tag{1}$$The experimentally verified near-Earth gravitational ... 3answers 988 views ### Validity of mean-field approximation In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where ... 1answer 2k views ### Can Hooke's law be derived? Can we derive Hooke's law from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations. 2answers 2k views ### Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration? The incompressible Navier-Stokes equations widely used in hydrodynamics don't have the gravitational acceleration.$$ \begin{align} \frac{\partial u_i}{\partial x_i} & = 0, \\ \frac{\partial u_i}{\... 4answers 2k views ### What does Feynman mean when he says thatF=ma$is not exact? Chapter 12-2 in Feynman Lectures Vol. 1 states: In fact the law,$F=ma$is not exactly true; if it were a definition we should have to say that it is always true; but it is not ... First, because ... 3answers 1k views ### Why do we still use perturbation theory, when we have advanced numerical methods and fast computers? If my question sounds ignorant or even insulting, I apologise. I may be completely wrong, since I'm not a theoretical physicist. So, I understand why perturbation theory was originally used in ... 1answer 2k views ### Why do physicists say that elementary particles are point particles? For example, an electron, it has mass and charge, but is considered to have point mass and point charge, but why? Why are they assumed to have charge and mass in a single infinitely small point in ... 1answer 431 views ### Hawking Radiation from the WKB Approximation Reading this paper which is itself an exposition of Parikh and Wilczek's paper, I get to a point where I fail to be able to follow the calculation. Now this is undoubtably because my calculational ... 1answer 612 views ### How to justify RPA (random phase approximation)? The Random Phase Approximation (RPA) is a technical method used in field theory to account for interactions when calculating correlation functions. It consists of only keeping a certain class of ... 2answers 917 views ### When is the adiabatic approximation for solid state systems valid? The adiabatic approximation for solid state systems is rather radical. I was wondering in which cases it breaks down. As it is based on the idea of the nuclii being much heavier than the electrons I ... 2answers 2k views ### What will be the equation of motion of driven pendulum for amplitudes beyond the small angle approximation? When finding the period of a pendulum beyond the small angle approximation, we have to use integration for small interval of$\theta$and elliptical integration. I was trying to apply this situation ... 1answer 1k views ### Why is that in the action principle, the Taylor's series is limited to the first order? For the Hamilton's principle: $$\delta s =\int_{t_1}^{t_2}L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) dt-\int_{t_1}^{t_2}L(\mathbf {q},\mathbf {\dot q},t) dt=0.\\$$ In the textbooks, ... 4answers 963 views ### Special Relativistic approximation to GR Some time ago I was talking to a professor in college about some of the fundamental aspects and origin of General Relativity. I was surprised to learn, in fact, that a pretty good approximation to GR ... 1answer 414 views ### A Problem With Deriving Ideal Gas Entropy From Multiplicity To derive the entropy of an ideal gas via the ergodic hypothesis, we first find the density of states function: $$g(E)=\frac{V^{N}}{h^{3N}}\frac{(2\pi m E)^{\frac{3N}{2}}}{\left(\frac{3N}{2}-1\right)!}... 7answers 32k views ### Do we take gravity = 9.8 m/s² for all heights when solving problems? Why or why not? Do we take gravity = 9.8 m/s² for all heights when solving problems? 2answers 694 views ### Approximating an expression for a potential In a problem which I was doing, I came across an expression for the potential V of a system as follows$$V = k\left(\frac{1}{l - x} + \frac{1}{l + x}\right)\tag{1}\label{1}$$where k is a constant,... 2answers 386 views ### How can we justify, in deriving quantum statistics, the use of Stirling approximation in the form \ln(x!)\approx x \ln x - x? At first sight one can say "why not to use only one term, or maybe three or more terms"? Why use two terms? I see that books (see for example good books like Griffiths quantum mechanics or Atkins ... 1answer 335 views ### Is drag force on an oscillating sphere an effective model for a swimmer? I saw the latest video from Sixty Symbols Little Swimmers. At the end of the videos he says that we do not know how to calculate the movement of the little swimmers. He says(6:14-6:40 in video) that ... 3answers 2k views ### Finding the energy eigenvalues of Hydrogen using WKB approach I need help to find the energy eigen values of Hydrogen atom using WKB approach. So far I know, the radial equation is given by$$\frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial ... 2answers 143 views ### Composite particles Dirac spinor approximation In many scattering process evolving composite particles such as proton, the composite particles are treated as an elementary particles. For example in electron proton scattering to the proton is ... 3answers 5k views ### Is Torricelli's law “wrong” for big holes? - Tank draining problem Consider a tank full of water with a constant cross-sectional area A1 placed vertically on the ground. Now someone drills a hole of an area A2 in the bottom of the tank, and the liquid starts escaping ... 1answer 176 views ### Does random phase approximation (RPA) response function obey Kramers-Kronig relations? Consider the screened Coulomb interaction in electron liquid, which in the random phase approximation (RPA) takes the form $$V(q,\omega)=\frac{v(q)}{1-v(q)\Pi(q,\omega)},$$ where$v(q)$is the ... 2answers 902 views ### WKB method of approximation Would it be legitimate to use the WKB approximation for a particle in a spherically symmetric Gaussian potential? $$V(r)~=~V_0(1-e^{-r^2/a^2}).$$ I'm not sure when to use which approximation method.... 1answer 227 views ### How to solve the 10000th eigenvalue of the anharmonic oscillator? Given a certain Hamiltonian, for example, $$H = -\frac{1}{2}\frac{\partial^2}{\partial x^2 } + x^4 .$$ , what methods can we use to approximate the$n$th eigenvalue, for very large$n$? For ... 1answer 341 views ### Rotating wave approximation and classical Rabi oscillations: why don't the fast oscillating terms seem negligible in the initial frame? I am trying to understand better the rotating wave approximation (RWA). Consider an atom modeled as a two level system, interacting with a Laser. I have the dipole momentum operator $$\vec{D} = d \... 1answer 251 views ### Eigenkets of degenerate perturbation theory Suppose the original Hamiltonian is H and we perturb it by a small potential V. The basis kets of the original hamiltonian H contains some degeneracy. Since there's some degeneracy, we take ... 1answer 1k views ### What is the range of validity of Fermi's Golden Rule? It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial ... 1answer 245 views ### A question on large-N limit? Let's take SU(N) for an example. The Lagrangian is$$\mathcal{L}=-\frac{1}{4g_{YM}^2}F_{\mu\nu}F^{\mu\nu}.$$We can define the t'Hooft coupling as$$\lambda=g_{YM}^2N.$$Then the large-N limit ... 1answer 130 views ### Approximate cloning of a qubit, given multiple starting copies Suppose I'm given several clones of a qubit in a pure unentangled state. That is to say, I'm given the state (a \left|0\right\rangle + b \left|1\right\rangle)^{\otimes n}. My goal is to make d ... 2answers 3k views ### Small oscillations of the double pendulum From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let m_2 be the lowest-hanging ... 2answers 281 views ### Approximations in simple pendulum In the approximation$$-(g/ \ell) \sin \theta \approx -(g/ \ell) \theta $$we make an error R which is O(\theta ^3). If i did well my calculations it is estimated by$$R\leq|(g / \ell)(\theta^3/3!)... 2answers 431 views ### Energy levels in close-proximity of each other in time-independent degenerate perturbation theory I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summarize this ... 1answer 372 views ### Why can we not apply perturbation theory in Born-Oppenheimer approximation In Weinberg's Lectures on Quantum Mechanics, he mentions Unfortunately, we cannot simply use first-order perturbation theory, with$T_{nuc}$taken as the perturbation and the state vectors$\Phi_{...
When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of ...
In Sakurai's book, the definition of differential cross section is: $$d\sigma/d\Omega= transition \;rate / probability\; flux$$ However this def doesn't contain any information about the thickness ...